Question

Sketch the vectors with the components \(\vec{A}=\left(A_{x}, A_{y}\right)=\) \((30.0 \mathrm{~m},-50.0 \mathrm{~m})\) and \(\vec{B}=\left(B_{x}, B_{y}\right)=(-30.0 \mathrm{~m}, 50.0 \mathrm{~m}),\) and find the magnitudes of these vectors.

Short answer

Question: Sketch the vectors A with components (30.0, -50.0) m and B with components (-30.0, 50.0) m, and find their magnitudes. Answer: The vectors A and B are sketched with components (30.0, -50.0) m and (-30.0, 50.0) m, respectively. The magnitudes of both vectors A and B are found to be 58.3 m.

Step-by-step solution

Reading the vector components

First, let's read the components of the two vectors A and B: - Vector A has components \(A_x = 30.0\) m and \(A_y = -50.0\) m, so vector A can be written as \(\vec{A} = (30.0, -50.0)\) m. - Vector B has components \(B_x = -30.0\) m and \(B_y = 50.0\) m, so vector B can be written as \(\vec{B} = (-30.0, 50.0)\) m.

Sketching the vectors

Using a Cartesian coordinate system, we can now sketch the two vectors with the given components. - To sketch \(\vec{A}\), start at the origin and draw an arrow that goes 30.0 m to the right and 50.0 m downward from the origin. This represents vector A with components \((30.0, -50.0)\) m. - To sketch \(\vec{B}\), start at the origin and draw an arrow that goes 30.0 m to the left and 50.0 m upward from the origin. This represents vector B with components \((-30.0, 50.0)\) m.

Finding the magnitudes of the vectors

Now we can find the magnitudes of the vectors using the formula \(\| \vec{A} \| = \sqrt{A_x^2 + A_y^2}\) for a vector \(\vec{A}=(A_x, A_y)\). - For vector \(\vec{A}\), the magnitude is: \(\| \vec{A} \| = \sqrt{(30.0 \mathrm{~m})^2 + (-50.0 \mathrm{~m})^2} = \sqrt{900 \mathrm{~m}^2 + 2500 \mathrm{~m}^2} = \sqrt{3400 \mathrm{~m}^2} = 58.3 \mathrm{~m}\) - Similarly, for vector \(\vec{B}\), the magnitude is: \(\| \vec{B} \| = \sqrt{(-30.0 \mathrm{~m})^2 + (50.0 \mathrm{~m})^2} = \sqrt{900 \mathrm{~m}^2 + 2500 \mathrm{~m}^2} = \sqrt{3400 \mathrm{~m}^2} = 58.3 \mathrm{~m}\) So, the magnitudes of vectors A and B are both 58.3 m.

Key concepts

Vector Components

Vector components are the individual parts that define a vector in a particular direction. Imagine a vector as an arrow pointing from one point to another. This arrow can be broken down into parts that describe its length and direction along the x-axis and y-axis (or any other coordinate system).
For instance, consider vector \(\vec{A}\) with components \(\(A_x = 30.0\) m\) and \(\(A_y = -50.0\) m\). Here, \(A_x\) represents how far along the x-axis the vector stretches, while \(A_y\) indicates the distance along the y-axis.

• These components allow us to understand the vector's behavior in a flat, two-dimensional space.
• Knowing vector components is crucial for operations like addition, subtraction, and calculating magnitudes.

Whenever you're dealing with vectors, it's important to clearly identify their components. Pay attention to the positive or negative signs as they signify the direction along each axis.

Vector Magnitude

The magnitude of a vector is a measure of its length, irrespective of its direction. It provides a sense of how "strong" or "long" the vector is. Imagine the vector as a line segment connecting two points. The vector magnitude is simply the length of that segment.
To find the magnitude of a vector, we use the Pythagorean theorem. If a vector \(\vec{A}\) is defined by its components \((A_x, A_y)\), then its magnitude \(\| \vec{A} \|\) is given by:\[\| \vec{A} \| = \sqrt{A_x^2 + A_y^2}\]

• This formula comes from considering the vector components as two sides of a right triangle, with the magnitude being the hypotenuse.
• For example, the magnitude of \(\vec{A} = (30.0, -50.0)\) m is calculated as \(\| \vec{A} \| = \sqrt{(30.0)^2 + (-50.0)^2} = 58.3\) m.

By understanding how to calculate the magnitude, you gain insights into the scale of the vector, which is essential for comparing different vectors, regardless of their direction.

Cartesian Coordinate System

The Cartesian coordinate system is a method for plotting points or vectors in space using a pair of numerical coordinates. These coordinates are defined by the perpendicular intersection of two lines, known as axes.

• The x-axis usually runs horizontally, while the y-axis runs vertically.
• The point where these axes intersect is the origin, denoted as \((0, 0)\).

In the system, any point or vector can be represented as \((x, y)\) where \(x\) is the distance along the horizontal axis and \(y\) is the distance along the vertical axis.
This coordinate system is essential for sketching vectors as it provides a framework to visualize their direction and magnitude. For instance, to sketch \(\vec{A} = (30.0, -50.0)\), you'd start at the origin, move 30.0 m to the right along the x-axis, and then 50.0 m downward along the y-axis.
Using a Cartesian coordinate system simplifies operations like vector addition and subtraction, as each operation can be performed separately on the x- and y-components.